# Set theoretic equation for Veronese varieties

Consider the embedding $$f:\mathbb{P}^n\rightarrow\mathbb{P}^N$$ induced by the complete linear system of degree $$d$$ hypersurfaces of $$\mathbb{P}^n$$. Its image $$V_{n,\,d}$$ is degree $$d$$ Veronese variety of dimension $$n$$.

The polynomials generating the ideal of $$V_{n,\, d}$$ are well-known: they are minors of a suitable matrix.

Question. Is it possible to cut-out $$V_{n, \, d}$$ set-theoretically with fewer equations?

For instance, we need three quadrics to generate the ideal of the twisted cubic $$V_{1,3}\subset\mathbb{P}^3$$, but $$V_{1,3}$$ is the set-theoretic intersection of a quadric and a cubic.

I am particularly interested in the case $$V_{2,\,3}\subset\mathbb{P}^9$$.

This is an attempt to address the special case of $$V_{2, 3} \subseteq \mathbb{P}^9$$. Denote the homogeneous coordinates on $$\mathbb{P}^9$$ by $$z_{ij}$$, $$i + j \leq 3$$, so that the embedding $$\mathbb{A}^2 \hookrightarrow V_{2, 3} \subseteq \mathbb{P}^9$$ is given by $$z_{ij} = x^iy^j$$. Then I believe the following $$7$$ equations cut $$V_{2, 3}$$ as a set theoretic complete intersection on $$\mathbb{P}^9$$: $$z_{20}z_{00} = z_{10}^2 \qquad z_{30}^2z_{00} = z_{20}^3 \qquad z_{02}z_{00} = z_{01}^2 \qquad z_{03}^2z_{00} = z_{02}^3$$ $$z_{12}z_{30} = z_{21}^2 \qquad z_{03}^2z_{30} = z_{12}^3 \qquad z_{11}^3 = z_{30}z_{03}z_{00}$$ (The first six equations are precisely the cubics and quadrics giving as set theoretic intersections the three twisted cubics coming from the "coordinate lines" on $$\mathbb{P}^2$$, and the last equation takes care of the remaining coordinate. An inductive procedure like this might work in general.)